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Chapter 2: Problem 64

Perform the operation and write the result in standard form. $$\frac{1+i}{i}-\frac{3}{4-i}$$

Short Answer

Expert verified
\(\frac{-5}{17} - \frac{20}{17}i\)

Step by step solution

01

Multiply the denominators in the fractions by their conjugates

The conjugate of \(i\) is \(-i\) and the conjugate of \(4-i\) is \(4+i\). So, multiply the first fraction by \(\frac{-i}{-i}\) to get \(\frac{1+i}{i} * \frac{-i}{-i} = \frac{-(1+i)}{1} = -1 - i\). For the second fraction, multiply by \(\frac{4+i}{4+i}\) to get: \(\frac{3}{4-i} * \frac{4+i}{4+i} = \frac{12+3i}{17} = \frac{12}{17} + \frac{3}{17}i\).
02

Subtract the second complex number from the first

Now subtract the two results from step 1. So, \((-1 - i) - (\frac{12}{17} + \frac{3}{17}i) = -1 - \frac{12}{17} - i - \frac{3}{17}i\). Simplify to get \(\frac{-5}{17} - \frac{20}{17}i\).
03

Write in standard form

The standard form for complex numbers is \(a + bi\). Here we already have our result in that form. So, \(\frac{-5}{17} - \frac{20}{17}i\) is the final answer.

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